# 1
# Portion of the circle x^2 + y^2 = 4 with x ≤ √2
# Let x== -2, the coordinate will look like (-2,0)
# Let x== √2, this coordinate will look like (√2,√2) or (√2,-√2)
t1 = var('t1')
circle1 = parametric_plot((2*cos(t1), 2*sin(t1)), (t1, pi/4, pi), color='blue')
circle2 = parametric_plot((2*cos(t1), 2*sin(t1)), (t1, pi, 7*pi/4), color='blue')

# Line segments
t=var('t')
line1 = parametric_plot((t, t), (t, 0, sqrt(2)), color='green')
line2 = parametric_plot((t, -t), (t, 0, sqrt(2)), color='green')
# Small circle centered at (0.5, 1.2)
small_circle = parametric_plot((0.1*cos(t) + 0.5, 0.1*sin(t) + 1.2), (t, 0, 2*pi), color='red')

P=(circle1 + circle2 + line1 + line2 + small_circle)
P

#2
# 1. Rotate by 90 degrees counter-clockwise
R = matrix([[0, -1], [1, 0]])

circle1_rotated = parametric_plot(R * vector((2*cos(t), 2*sin(t))), (t, pi/4, pi), color='blue')
circle2_rotated = parametric_plot(R * vector((2*cos(t), 2*sin(t))), (t, pi, 7*pi/4), color='blue')
line1_rotated = parametric_plot(R * vector((t, t)), (t, 0, sqrt(2)), color='green')
line2_rotated = parametric_plot(R * vector((t, -t)), (t, 0, sqrt(2)), color='green')
small_circle_rotated = parametric_plot(R * vector((0.1*cos(t) + 0.5, 0.1*sin(t) + 1.2)), (t, 0, 2*pi), color='red')

P_rotated = circle1_rotated + circle2_rotated + line1_rotated + line2_rotated + small_circle_rotated
show(P_rotated)

# 2. Stretch horizontally and compress vertically by a factor of 2
S = matrix([[2, 0], [0, 1/2]])

circle1_stretched = parametric_plot(S * vector((2*cos(t), 2*sin(t))), (t, pi/4, pi), color='blue')
circle2_stretched = parametric_plot(S * vector((2*cos(t), 2*sin(t))), (t, pi, 7*pi/4), color='blue')
line1_stretched = parametric_plot(S * vector((t, t)), (t, 0, sqrt(2)), color='green')
line2_stretched = parametric_plot(S * vector((t, -t)), (t, 0, sqrt(2)), color='green')
small_circle_stretched = parametric_plot(S * vector((0.1*cos(t) + 0.5, 0.1*sin(t) + 1.2)), (t, 0, 2*pi), color='red')

P_stretched = circle1_stretched + circle2_stretched + line1_stretched + line2_stretched + small_circle_stretched
show(P_stretched)

# 3. Reflect through line y = -x
M = matrix([[0, -1], [-1, 0]])

circle1_reflected = parametric_plot(M * vector((2*cos(t), 2*sin(t))), (t, pi/4, pi), color='blue')
circle2_reflected = parametric_plot(M * vector((2*cos(t), 2*sin(t))), (t, pi, 7*pi/4), color='blue')
line1_reflected = parametric_plot(M * vector((t, t)), (t, 0, sqrt(2)), color='green')
line2_reflected = parametric_plot(M * vector((t, -t)), (t, 0, sqrt(2)), color='green')
small_circle_reflected = parametric_plot(M * vector((0.1*cos(t) + 0.5, 0.1*sin(t) + 1.2)), (t, 0, 2*pi), color='red')

P_reflected = circle1_reflected + circle2_reflected + line1_reflected + line2_reflected + small_circle_reflected
show(P_reflected)

# 4. Project onto the line y = -2x
P_matrix = matrix([[1/5, -2/5], [-2/5, 4/5]])

circle1_projected = parametric_plot(P_matrix * vector((2*cos(t), 2*sin(t))), (t, pi/4, pi), color='blue')
circle2_projected = parametric_plot(P_matrix * vector((2*cos(t), 2*sin(t))), (t, pi, 7*pi/4), color='blue')
line1_projected = parametric_plot(P_matrix * vector((t, t)), (t, 0, sqrt(2)), color='green')
line2_projected = parametric_plot(P_matrix * vector((t, -t)), (t, 0, sqrt(2)), color='green')
small_circle_projected = parametric_plot(P_matrix * vector((0.1*cos(t) + 0.5, 0.1*sin(t) + 1.2)), (t, 0, 2*pi), color='red')

P_projected = circle1_projected + circle2_projected + line1_projected + line2_projected + small_circle_projected
show(P_projected)

# 3
B = Matrix([[1, 2], [0, 1]]) 
theta = pi / 4 
A = Matrix([[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]) 

# Define the standard basis vectors
e1 = vector([1, 0])
e2 = vector([0, 1])
# Apply the shearing transformation
Te1=A*e1
Te2=A*e2
Te1
Te2
# Apply the rotation transformation on the sheared vectors
STe1=B*Te1
STe2=B*Te2
STe1
STe2

# Create a plot for the original and transformed vectors using lines
(plot(e1,color='blue')+plot(e2,color='blue',aspect_ratio=1)+plot(Te1,color='red')+plot(Te2,color='red',aspect_ratio=1)+plot(B*A*e1,color='green')+plot(B*A*e2,color='green',aspect_ratio=1)+parametric_plot( (.5*cos(t),.5*sin(t)),(t,0,pi/3),color='black')+plot(x,(x,-1,1),linestyle='--'))

# Calculate composite matrix S(T(x)) = S ◦ T
composite_matrix = B * A
# Compute products of standard matrices AB and BA
AB = A * B
BA = B * A
composite_matrix, AB, BA

# 4)
# The two compositions involve different sequences of transformations, leading to different final results. Due to the differing effects of shearing and rotation p (S ◦ T)(x) is not equal to S(T(x)).

# 5
# Define the matrices
A1 = Matrix([[1, 0], [0, 1], [0, 0]])
A2 = Matrix([[1, 2], [3, -1], [2, 0]])
A3 = Matrix([[1, 2], [-1, -2], [3, -3]])

# Set the range for x and y
x_range = (-10, 10)
y_range = (-10, 10)

# Create the parametric plots for A1
x, y = var('x y')
X1 = A1[0, 0]*x + A1[0, 1]*y
Y1 = A1[1, 0]*x + A1[1, 1]*y
Z1 = A1[2, 0]*x + A1[2, 1]*y
plot_A1 = parametric_plot3d((X1, Y1, Z1), (x, x_range[0], x_range[1]), (y, y_range[0], y_range[1]),color="red")

# Create the parametric plots for A2
X2 = A2[0, 0]*x + A2[0, 1]*y
Y2 = A2[1, 0]*x + A2[1, 1]*y
Z2 = A2[2, 0]*x + A2[2, 1]*y
plot_A2 = parametric_plot3d((X2, Y2, Z2), (x, x_range[0], x_range[1]), (y, y_range[0], y_range[1]),color="green")

# Create the parametric plots for A3
X3 = A3[0, 0]*x + A3[0, 1]*y
Y3 = A3[1, 0]*x + A3[1, 1]*y
Z3 = A3[2, 0]*x + A3[2, 1]*y
plot_A3 = parametric_plot3d((X3, Y3, Z3), (x, x_range[0], x_range[1]), (y, y_range[0], y_range[1]))

# Show the plots
show(plot_A1 + plot_A2 + plot_A3)

# Define the parameters
s, t = var('s t')

# Define the parametric equations for the paraboloid
X = s * cos(t)
Y = s * sin(t)
Z = s^2

# Create the 3D parametric plot for the paraboloid
plot_G = parametric_plot3d((X, Y, Z), (s, 0, 2), (t, 0, 2*pi))

# Show the plot
show(plot_G)

# Define the parameters
s, t = var('s t')

# Define the parametric equations for the paraboloid
X = s * cos(t)
Y = s * sin(t)
Z = s^2

# Create the parametric plot for the paraboloid
plot_G = parametric_plot3d((X, Y, Z), (s, 0, 2), (t, 0, 2*pi), color='lightgray')

# Define the matrices
A4 = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
A5 = Matrix([[1, 0, 1], [0, 1, 1], [1, 1, 0]])
A6 = Matrix([[-1, 0, 1], [0, 1, -1], [1, -1, 0]])
A7 = Matrix([[1, -1, 1], [2, -2, 2], [-1, 1, -1]])

# Create the transformed plots for A4
plot_A4 = parametric_plot3d(
    (A4[0, 0] * X + A4[0, 1] * Y + A4[0, 2] * Z, 
     A4[1, 0] * X + A4[1, 1] * Y + A4[1, 2] * Z, 
     A4[2, 0] * X + A4[2, 1] * Y + A4[2, 2] * Z),
    (s, 0, 2), (t, 0, 2 * pi), color='blue'
)

# Create the transformed plots for A5
plot_A5 = parametric_plot3d(
    (A5[0, 0] * X + A5[0, 1] * Y + A5[0, 2] * Z, 
     A5[1, 0] * X + A5[1, 1] * Y + A5[1, 2] * Z, 
     A5[2, 0] * X + A5[2, 1] * Y + A5[2, 2] * Z),
    (s, 0, 2), (t, 0, 2 * pi), color='red')

# Create the transformed plots for A6
plot_A6 = parametric_plot3d(
    (A6[0, 0] * X + A6[0, 1] * Y + A6[0, 2] * Z, 
     A6[1, 0] * X + A6[1, 1] * Y + A6[1, 2] * Z, 
     A6[2, 0] * X + A6[2, 1] * Y + A6[2, 2] * Z),
    (s, 0, 2), (t, 0, 2 * pi), color='green'
)

# Create the transformed plots for A7
plot_A7 = parametric_plot3d(
    (A7[0, 0] * X + A7[0, 1] * Y + A7[0, 2] * Z, 
     A7[1, 0] * X + A7[1, 1] * Y + A7[1, 2] * Z, 
     A7[2, 0] * X + A7[2, 1] * Y + A7[2, 2] * Z),
    (s, 0, 2), (t, 0, 2 * pi), color='purple'
)

# Show all plots together
show(plot_G + plot_A4 + plot_A5 + plot_A6 + plot_A7)

# Define the standard basis vectors
e1 = vector([1, 0, 0])
e2 = vector([0, 1, 0])
e3 = vector([0, 0, 1])

# Create a grid of points based on linear combinations of e1, e2, e3
grid_points = []
for x in range(-2, 2):
    for y in range(-2, 2):
        for z in range(-2, 2): 
            grid_points.append(x * e1 + y * e2 + z * e3)

# Create the parametric plot for the grid
plot_H = point3d(grid_points, color='lightgray', size=30)

# Define the matrices
A4 = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
A5 = Matrix([[1, 0, 1], [0, 1, 1], [1, 1, 0]])
A6 = Matrix([[-1, 0, 1], [0, 1, -1], [1, -1, 0]])
A7 = Matrix([[1, -1, 1], [2, -2, 2], [-1, 1, -1]])

# Transform grid points for each matrix
transformed_A4 = [A4 * p for p in grid_points]
transformed_A5 = [A5 * p for p in grid_points]
transformed_A6 = [A6 * p for p in grid_points]
transformed_A7 = [A7 * p for p in grid_points]

# Create the transformed plots
plot_T_A4 = point3d(transformed_A4, color='blue', size=30)
plot_T_A5 = point3d(transformed_A5, color='red', size=30)
plot_T_A6 = point3d(transformed_A6, color='green', size=30)
plot_T_A7 = point3d(transformed_A7, color='purple', size=30)

# Show all plots together
show(plot_H + plot_T_A4 + plot_T_A5 + plot_T_A6 + plot_T_A7)